In this specification, the following terms have the following definitions as given alongside. These are additions to the usual definitions expressed in the art.
Extrusion is a process used to create objects of a fixed cross-sectional profile in which a material is pushed or drawn through a die of a desired cross-section.
Die is a specialized tool used in the manufacturing industry to shape or cut polymeric materials, metallic materials and the like.
Die swell is the phenomenon by which a polymer after exiting from a die partially recovers or ‘swells back’ towards the former shape and volume. Die swell occurs when a polymer melt gains uniform velocity across the section after exiting from the die.
Die geometry is the dimensional specification and structure of the die.
Die parameters include die entrance angle, cross-sectional area, length and material of the die.
ALE (Arbitrary Lagrangian Eulerian) framework is a combination of Lagrangian and Eulerian methods implemented for simplifying and solving partial differential equations representing the behavior of any system.
Finite Element Method (FEM) is a numerical technique for finding approximate solutions of Partial Differential Equations (PDE). The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta and the like.
Mesh is a basic and standard entity used for obtaining numerical solutions of partial differential equations through the finite element method representing the volume/area of the domain of analysis.
Computational mesh geometry represents the volume/area of the domain of analysis.
Extrudate geometry represents the structural specification of the polymer extrudate.
Polymer melt parameters include molecular weight, branching, viscosity and elasticity of the polymer.
Rheological characterization is the experimental characterization of a material's rheological behaviour. Rheology is the relation of the flow or deformation behaviour of a material and in reaction to applied external forces.
Constitutive equation is a relation between two physical quantities which are specific to a material or substance, and approximates the response of that material to external forces. It is combined with other equations governing physical laws to solve physical problems, like the flow of a fluid in a pipe, or the response of a crystal to an electric field.
Polymer melt motion represents the motion of the polymer melt material through the die which is characterized by flow velocity, pressure, stress and the like.
Mesh motion represents the change in the 3-dimensional co-ordinates of the vertices of the computational mesh cells.
A boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Pharr-Thien Tanner (PTT) model is a simple model used to simulate the rheological behaviour of polymer melts and concentrated solutions. The PTT model is derived from the network theory proposed by Phan-Thien and Tanner in 1977. The PTT model has found widespread use in numerical simulations of the flow of polymer solutions and melts. It is shown to be an excellent simple differential model for the elongational properties of polymer solutions (as published in the journal of International Congress on Rheology, Cambridge, UK, 2000 in the article-'Phan-Thien Tanner flow in concentric annuli').
Godunov update scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. In this method, the conservative variables are considered as piecewise constant over the mesh cells at each time step and the time evolution is determined by the exact solution of the Riemann problem at the inter-cell boundaries (as established by Hirsch, 1990). A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with a piecewise constant data having a single discontinuity.
Lagrangian motion represents the free surface motion of polymer melt simulations in Lagrangian mode. Lagrangian framework is a differential equation whose solutions are the functions for which a given functional is stationary. The simpler representation of the polymer constitutive equations in a Lagrangian framework typically avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation.
Through the polymer melt extrusion process, very complex cross-sections of polymeric plastics can be created. Finished parts can be formed with an excellent surface finish. Extrusion can be continuous (theoretically producing indefinitely long material) or semi-continuous (producing many pieces). The polymer extrusion process is done in a molten state.
Polymer melt extrusion processes are widely used in the synthetic polymer industries and involve flow of viscoelastic polymeric fluids through complex flow geometries. Viscoelasticity is the property of materials that exhibits both viscous and elastic characteristics when undergoing deformation. Viscous materials, like synthetic and natural polymers, resist shear flow and develop strain with time when a stress is applied. Elastic materials strain instantaneously when stretched and quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and exhibit time dependent strain.
Die swell, also known as an extrudate swell, is a common phenomenon in polymer processing. The swelling of polymer extrudate is due to a fading memory effect of viscoelastic polymer melt. Die swell occurs in instances of polymer extrusion, in which a stream of polymeric melt is forced through a die. Die swell is a phenomenon directly related to entropy and the relaxation of the polymer within the flow stream after exiting from the die when external constraints to flow cease to exist. When polymer melt is extruded, it is subjected to an increased flow rate, due to which the polymer chains get stretched. Physical entanglements may relax, if the time scale of the polymer within the die is long enough. When the polymer stream leaves the die, the remaining physical entanglements cause the polymers in the die stream to regain a portion of its former shape by relaxation of the stretched chains, in order to return to the original conformation that minimizes the entropy.
Computational calculation of the die swell of the polymer melt in an extrusion process is important for appropriate die design in profile extrusion applications. This is a challenging task due to the requirement for simulation of the free surface which needs special techniques in the traditionally used Eulerian framework. The degree of swell depends on the coupling between the shear and extensional viscosity of the polymer melt which is modelled by complex constitutive equations having convective derivatives of stress.
Hence it is appropriate to have a better understanding of this flow behavior of the polymer melt and resulting swell which can ultimately lead to improvements in the understanding of the extrusion process, for optimization of both the die design and the processing parameters. Therefore, there is a need to have a tool and a method for simulation of the die swell using a technique which provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion.
Existing Knowledge:
Most of the numerical studies on die swell reported are based on the Streamline Finite Element Method (SFEM) introduced by Luo and Tanner [as published in ‘A streamline element scheme for solving viscoelastic flow problems. Part II. Integral constitutive models’, J. Non-Newtonian Fluid Mech. 22 (1986) 61-89] which offers a simple algorithm for solving integral constitutive equations and is suitable for the extrusion operation.
SFEM was subsequently modified by Luo and Mitsoulis [as published in ‘An efficient algorithm for strain history tracking in finite element computation of Non-Newtonian fluids with integral constitutive equations’, Int. J. Numer. Meth. Fluids. 11 (1990) 1015-1031], who introduced a particle-tracking scheme on the streamline by using Picard iterative scheme which decouples the computation of the free surface shape from that of velocity and stress fields. They used the iterative method of Dupont and Crochet [as published in ‘The vortex growth of a KBKZ fluid in an abrupt contraction’, J. Non-Newtonian Fluid Mech., 29 (1988) 81] in which the strain history of the material particles was calculated along streamlines.
Several features of the earlier numerical method have been modified by Goublomme et al. [as published in ‘Numerical prediction of extrudate swell of a high-density polyethylene’, J. Non-Newtonian Fluid Mech. 44 (1992) 171-195] to simulate at high shear rates by introducing a fourth-order Runge-Kutta algorithm to calculate the path lines and the strains within the parent element. The iterative algorithm was of the incremental loading type, where a numerical parameter controls the transformation of a Newtonian solution into a viscoelastic one. The algorithm developed by Goublomme et al. to simulate swell is used in the commercial software POLYFLOW®.
A few researchers have employed the Newton iterative scheme to compute the free surface shape simultaneously with velocity and stress values (coupled method). Be'raudo et al. [as published in ‘A finite element method for computing the flow of multi-mode viscoelastic fluids: comparison with experiments’, J. Non-Newtonian Fluid Mech. 75 (1998) 1-23] used the finite element method based on Newton's iterative scheme and on discontinuous approximations of the extra-stress tensor. The discontinuous interpolation eliminated the stress variable at the element level by means of a static condensation technique. Simple computation of the stream function which allows the determination of the free surface was used.
All these methods were based on the finite element technique in the Eulerian framework using streamlines/steam tube to compute the swell. The free surface was computed by the predictor corrector technique based on the streamline method and only steady state die swell simulation was possible.
The die swell simulations in Lagrangian framework had an advantage since the surface evolves naturally with the material flow. But there were problems associated with mesh distortion and frequent remeshing was required.
There are patents and patent applications which disclose methods of analysis and correction of die swell.
For instance, U.S. Pat. No. 5,874,034 discloses a method and apparatus for extruding viscoelastic polymers. This patent discloses an extrusion die for manufacturing viscoelastic and similar polymers while reducing or eliminating the die swell. The method and apparatus can be used to manufacture a xerographic toner that has a relatively low melting temperature, which can cause a reduction in the power consumption of a printer or copier. Use of toner particles created using the disclosed technique can reduce or eliminate the phenomenon known as “vinyl off set,” which causes the transfer of a hard copy image onto a vinyl material. The method disclosed in this patent does not provide a reliable method to calculate the die swell. The reduction of die swell is dependent on the toner particles and there is no correction mechanism for the die swell caused by any characteristic change of the toner particle.
Similarly, PCT application W00015408 discloses a system and a method for generating die swell/draw down information for a profile extrusion die design. The system employs a rheometer; a slit die and puller mechanism to obtain the die swell/draw down information and a processor which uses this information to generate a die design chart which will help in the design of a profile extrusion die. The drawback of the disclosed method is that any real-time dynamic die swell correction is not possible.
Further, U.S. Pat. No. 7,110,921 discloses a method for designing a profile extrusion based on the analogy that exists between membrane deflection under pressure and low Reynolds number pressure driven fluid flow. The relative shape of the die is predicted using the analogy for die swell correction. A drawback of the disclosed method is that the analogy is not strong enough to give accurate die swell predictions for all polymer materials and flow rate as the disclosed method can only be used for low Reynolds number pressure driven fluid flow.
Finally, US Patent application 20060223961 discloses a method for selecting a polyolefin having a die swell within a pre-selected range. The method includes i) obtaining a data set for at least one property from plurality of polyolefin batches of reference samples, other than die swell, of the polyolefin; ii) obtaining a die swell data set from the reference samples of the polyolefin; iii) correlating at least one property and the die swell data set using a regression analysis to generate an equation for predicting the die swell as a function of at least one property. The die swell prediction is not authenticated as it is based only on a single property. The die swell is affected by a plurality of parameters of the die and the polymer melt.
Therefore, there is a need for:                a simple and reliable system to analyze die swell;        a system to calculate accurate die swell taking all parameters of the die and polymer melt that can affect the die swell into consideration;        a system to calculate and correct the die swell dynamically; and        a system to store the die swell corresponding to the predetermined die and melt parameters and profiles.        